The known orthogonality relations for the factors \(\varphi _\lambda \)  (which are the normalized characters of the
symmetric group in representation \(\lambda \) ) [1]} are as follows:
\(\zeta _{\Delta } \sum _{\lambda } \left(\frac{{\rm \dim }\lambda }{d!}\right)^2\varphi _\lambda (\mu )\varphi _\lambda (\Delta ) =\delta _{\Delta ,\mu }\) 
